Indirect Standardization and SMR

Indirect standardization is used to calculate the expected mortality rate for the index population, given age specific mortality rates from a reference population.

The method applies not only to mortality rates but also to any rates of uncommon events (i.e. the Poisson distribution can be applied). An example of a different application is calculation of standardized incidence rate ratio (SIR) for a disease in a population.

If you want to standardize by both age and sex then enter two sets of age groups (i.e. 10 rows instead of 5 for the example below) split into male and female consecutively. Provided that you have sex and age specific mortality rates for the reference population and age and sex specific population sizes for the index population then you can produce an age and sex standardized SMR.

SMRs from different index/study populations are not strictly comparable because they are calculated using different weighting schemes that depend upon the age structures of the index/study populations. SMRs can, however, be compared if you make the assumption that the ratio of rates between index and reference populations is constant; this is similar to the assumption of proportional hazards in Cox regression (Armitage and Berry, 1994). Direct standardization is an alternative to indirect standardization that does provide comparable measures. Direct standardization, however, has other weaknesses, such as greater susceptibility than the indirect method to error with small numbers, which make the choice of method a matter for careful statistical consideration. If you are in doubt, please consult with a Statistician and/or Epidemiologist.

Please note that methods of standardization can mask differences in rates between populations; in order to avoid this you should supplement SMR analysis with individual comparisons of group (e.g. age) specific rates.

Data input

Groups/strata, e.g. age bands for indirect age standardization

Rates for each group from a reference population

Person-time for each group from the index/study population (e.g. size of each group if just one year and all subjects were followed up)

Multiplier for reference rates, e.g. 10000 if mortality entered as deaths per 10000, 1 if mortality entered as a decimal fraction

The SMR is expressed in ratio and integer (ratio * 100) formats with a confidence interval.

The confidence intervals are calculated by the exact Poisson method of Owen, this gives better coverage than the frequently quoted Vandenbroucke approximation or other asymptotic methods (Ulm, 1990; Greenland, 1990).

- where SMRl and SMRu are lower and upper confidence limits respectively for the SMR, χ2α, ν is the (100*α)th chi-square centile with ν degrees of freedom, d is the number of observed deaths, e is the number of expected deaths, ni is the person-time for the ith study group stratum and Ri is the reference population rate for the ith stratum.

The following data represent the age-specific mortality rates for liver cirrhosis in men and the number of male doctors in each age stratum:

Age group

Mortality per million men per year

Number of male doctors

15-24

5.859

1080

25-34

13.050

12860

35-44

46.937

11510

45-54

161.503

10330

55-64

271.358

7790

To analyse these data in StatsDirect you must select indirect standardization and SMR from the rates section of the analysis menu. Enter the mortality rate and group size for each age group. Note that group size refers to the study sample of doctors and not to the male population used to derive mortality data. Enter the mortality denominator as 1000000. Enter the observed deaths as 14.

For this example:

Indirectly Standardized Rates and SMR

Reference rate

Observed person-time

Expected deaths

0.000006

1080

0.006328

0.000013

12860

0.167823

0.000047

11510

0.540245

0.000162

10330

1.668326

0.000271

7790

2.113879

Total = 4.4966004

Standardized Mortality Ratio (SMR) = 3.113463

SMR (*100 as integer) = 311

Exact 95% confidence interval = 1.702159 to 5.223862 (170 to 522)

Probability of observing 14 or more deaths by chance P = .0002

Probability of observing 14 or fewer deaths by chance P > .9999

Here we can see that the total expected deaths from liver cirrhosis in male doctors is 4.5 per year. The observed number, 14, was statistically highly significantly greater than expected. With 95% confidence we can state that male doctors in this country exhibit between 1.7 and 5.2 times the number of deaths from liver cirrhosis than expected from the general male population of a similar age distribution.